Noncommutative Algebra and Applications

Projeto Temático FAPESP No.2015/09162-9, coordenado por César Polcino Milies

Research group seminars

2019 seminars:

Lecturer	Title	Date	Time	Room
Pedro Russo (IME-USP)	On the existence of free noncyclic groups in finite dimensional division rings with involution	November 05, 2019	14:30-15:30	242-A
Abstract: Lichtman has conjectured that a non central normal subgroup N of the multiplicative group of a division ring D contains a non cyclic free subgroup. We address the particular case where D is finite dimensional over its center k, which is supposed to be uncountable and whose characteristic is different from 2, and is endowed with a k-involution. Under these circumstances, if N contains the nonzero members of k, then it contains a free non cyclic subgroup whose free generators are symmetric.

Lecturer	Title	Date	Time	Room
Jairo Z. Gonçalves (IME-USP)	Constructing free and free symmetric pairs in cyclic algebras with an involution	October 29, 2019	14:30-15:30	242-A
Abstract: Let A = (K/F, \sigma, Y, b) be a cyclic algebra, where K/F is a Galois extension over the non absolute field F with Galois group < \sigma> of order n, and let b \in F. Let us assume, moreover, that A=(K/F, \sigma, Y, b) satisfies the relations Y^n = b and YaY^{-1}=a^{\sigma} , for all a \in K. Let U=U(A) be the group of units of A. We show how to construct pairs (u, v) in U such that <u, v> is a free group of rank two. Also, if char F \neq 2, * is an F-involution of A such that K*=K, and if A is not the quaternion algebra with the simplectic involution, then we exhibit a pair (u, v) in U(A) such that u* = u, v* = v and <u, v> is a free group of rank two.

Lecturer	Title	Date	Time	Room
Tran Giang Nam (Institute of Mathematics, VAST, Vietnam)	Realizing corners of Leavitt path algebras as Steinberg algebras, with corresponding connections to graph C*-algebras (part 2)	October 22, 2019	14:30-15:30	242-A
Abstract: We show that the endomorphism ring of any nonzero finitely generated projective module over the Leavitt path algebra L_K(E) of an arbitrary graph E with coefficients in a field K is isomorphic to a Steinberg algebra. This yields in particular that every nonzero corner of the Leavitt path algebra of an arbitrary graph is isomorphic to a Steinberg algebra. This in its turn gives that every K-algebra with local units which is Morita equivalent to the Leavitt path algebra of a row-countable graph is  isomorphic to a Steinberg algebra. Moreover, we prove that a corner by a projection of a C*-algebra of a countable graph is isomorphic to the C*-algebra of an ample groupoid. (Joint work with Gene Abrams and Misha Dokuchaev.)

Lecturer	Title	Date	Time	Room
Tran Giang Nam (Institute of Mathematics, VAST, Vietnam)	Realizing corners of Leavitt path algebras as Steinberg algebras, with corresponding connections to graph C*-algebras	October 08, 2019	14:30-15:30	242-A
Abstract: We show that the endomorphism ring of any nonzero finitely generated projective module over the Leavitt path algebra L_K(E) of an arbitrary graph E with coefficients in a field K is isomorphic to a Steinberg algebra. This yields in particular that every nonzero corner of the Leavitt path algebra of an arbitrary graph is isomorphic to a Steinberg algebra. This in its turn gives that every K-algebra with local units which is Morita equivalent to the Leavitt path algebra of a row-countable graph is  isomorphic to a Steinberg algebra. Moreover, we prove that a corner by a projection of a C*-algebra of a countable graph is isomorphic to the C*-algebra of an ample groupoid. (Joint work with Gene Abrams and Misha Dokuchaev.)

Lecturer	Title	Date	Time	Room
André Pereira (UFRRJ)	Algebras of p-groups of class 2 over the rationals	October 01, 2019	14:30-15:30	242-A
Abstract: The goal of this presentation is to analyze p-groups, M and N, with nilpotency class 2 and same rational group algebras. We prove that if QM = QN, then their commutator subgroups are equal and the centers, Z(M) and Z(N), have their orders preserved. After this, we apply the result to Frattini Central p- group, this is, the Frattini subgroup is in the center of the group. To finish, we present an example of two p-groups of order p^7, with nilpotency class 2 such that they have the same rational group algebra, but the centers of these groups are not isomorphic.

Lecturer	Title	Date	Time	Room
Raul Ferraz (IME-USP)	One weight codes in some classes of group rings	September 17, 2019	14:30-15:30	242-A
Abstract: Let F_q be a finite field with q elements and G be a group of order n. Firstly we give conditions to ensure that a cyclic code in F_q G is a  one-weight code in the semi simple case (that is when gcd(n,q) = 1). Further we consider the case when G is abelian, and also when gcd(n,q) >1. (Joint work with Ruth Nascimento Ferreira from Universidade Tecnológica Federal do Paraná, Guarapuava-PR.)

Lecturer	Title	Date	Time	Room
Ángel del Río (University of Murcia, Spain)	The Zassenhaus Conjecture for cyclic-by-abelian groups, with some proofs	September 10, 2019	14:30-15:30	242-A
Abstract: Let G be a finite group. The Zassenhaus conjecture states that every torsion unit of augmentation 1 in the integral group algebra is conjugate in the rational group algebra to an element of G. Although a counterexample has been found recently in the class of metabelian groups, it has been proved for same large classes of groups including nilpotent groups and cyclic-by-abelian groups, and it is open for the class of supersolvable groups. We will present some positive results for the class of cyclic-by-nilpotent groups obtained recently in cooperation with Mauricio Caicedo.

Group Rings, Groups and Rings

São Paulo, September 2 to 5, 2019

In honor of Professor Jairo Z. Gonçalves

on the occasion of his 70th birthday

Lecturer	Title	Date	Time	Room
Mayumi Makuta (IME-USP)	Obstructions to extensions of semilattices of groups by groups (part 2)	August, 20, 2019	14:30-15:30	242-A
Abstract: In this talk, we give an interpretation of the third partial cohomology group (defined by Dokuchaev and Khrypchenko) with values in an abelian semilattice of groups in terms of extensions of a semilattice of groups by a group. For thisend, we define a partial abstract kernel inspired on the classic case of group cohomology and the case of inverse semigroup cohomology.

Lecturer	Title	Date	Time	Room
Mayumi Makuta (IME-USP)	Obstructions to extensions of semilattices of groups by groups	August, 13, 2019	14:30-15:30	242-A
Abstract: In this talk, we give an interpretation of the third partial cohomology group (defined by Dokuchaev and Khrypchenko) with values in an abelian semilattice of groups in terms of extensions of a semilattice of groups by a group. For thisend, we define a partial abstract kernel inspired on the classic case of group cohomology and the case of inverse semigroup cohomology.

Lecturer	Title	Date	Time	Room
Jairo Z. Gonçalves (IME-USP)	Free pairs of symmetric elements in normal subgroups of division rings (part 2)	June 11, 2019	14:30-15:30	242-A
Abstract: Let D be a division ring with central subfield k of characteristic different from 2, let * be a k-involution of D, and let N be a normal subgroup of the multiplicative group of D. We show that if G \subseteq N is a *-stable nonabelian subgroup that is either torsion-free polycyclic-by-finite but not abelian-by-finite, or finite of odd order, then N contains a pair (u, v) of elements such that u^*=u, v^*=v and such that \langle u, v \rangle is a noncyclic free group. One aspect of the above proof requires that we extend a theorem of Bergman on invariant ideals in commutative group algebras. This new result is surely of interest in its own right. It appears in the Appendix and can be read independently of the remainder of the paper.

Lecturer	Title	Date	Time	Room
Jairo Z. Gonçalves (IME-USP)	Free pairs of symmetric elements in normal subgroups of division rings	June 04, 2019	14:30-15:30	242-A
Abstract: Let D be a division ring with central subfield k of characteristic different from 2, let * be a k-involution of D, and let N be a normal subgroup of the multiplicative group of D. We show that if G \subseteq N is a *-stable nonabelian subgroup that is either torsion-free polycyclic-by-finite but not abelian-by-finite, or finite of odd order, then N contains a pair (u, v) of elements such that u^*=u, v^*=v and such that \langle u, v \rangle is a noncyclic free group. One aspect of the above proof requires that we extend a theorem of Bergman on invariant ideals in commutative group algebras. This new result is surely of interest in its own right. It appears in the Appendix and can be read independently of the remainder of the paper.

Lecturer	Title	Date	Time	Room
Mikhailo Dokuchaev (IME-USP)	A Chase-Harrison-Rosenberg sequence for a partial Galois extension of commutative rings (part 3)	May 28, 2019	14:30-15:30	242-A
Abstract: In a paper published in 1965 S.U. Chase, D.K. Harrison and A. Rosenberg obtained a series of results on Galois extensions of commutative rings, including several equivalent definitions, a Galois correspondence and a seven term exact sequence, involving Galois cohomology groups, Picard groups and the relative Brauer group. The sequence is seen as a common generalization of two classical facts on Galois extensions of fields: the Hilbert's Theorem 90 and the isomorphism of the relative Brauer group with the second Galois cohomology group. In a joint paper with M. Ferrero and A. Paques (2007) we defined Galois extensions in the context of partial group actions and obtained a Galois correspondence. It is natural to extend the Chase-Harrison-Rosenberg sequence for the case of partial Galois extensions of commutative rings. In a joint paper with A. Paques and H. Pinedo we built up homomorphisms, which are analogues of those in the Chase-Harrison-Rosenberg sequence. Now in a preprint with A. Paques, H. Pinedo and I. Rocha we prove that the sequence is exact. We shall recall the ingredients of the sequence and discuss some ideas used in the proof of the exactness.

Lecturer	Title	Date	Time	Room
Mikhailo Dokuchaev (IME-USP)	A Chase-Harrison-Rosenberg sequence for a partial Galois extension of commutative rings (part 2)	May 21, 2019	14:30-15:30	242-A
Abstract: In a paper published in 1965 S.U. Chase, D.K. Harrison and A. Rosenberg obtained a series of results on Galois extensions of commutative rings, including several equivalent definitions, a Galois correspondence and a seven term exact sequence, involving Galois cohomology groups, Picard groups and the relative Brauer group. The sequence is seen as a common generalization of two classical facts on Galois extensions of fields: the Hilbert's Theorem 90 and the isomorphism of the relative Brauer group with the second Galois cohomology group. In a joint paper with M. Ferrero and A. Paques (2007) we defined Galois extensions in the context of partial group actions and obtained a Galois correspondence. It is natural to extend the Chase-Harrison-Rosenberg sequence for the case of partial Galois extensions of commutative rings. In a joint paper with A. Paques and H. Pinedo we built up homomorphisms, which are analogues of those in the Chase-Harrison-Rosenberg sequence. Now in a preprint with A. Paques, H. Pinedo and I. Rocha we prove that the sequence is exact. We shall recall the ingredients of the sequence and discuss some ideas used in the proof of the exactness.

Lecturer	Title	Date	Time	Room
Mikhailo Dokuchaev (IME-USP)	A Chase-Harrison-Rosenberg sequence for a partial Galois extension of commutative rings	May 07, 2019	14:30-15:30	242-A
Abstract: In a paper published in 1965 S.U. Chase, D.K. Harrison and A. Rosenberg obtained a series of results on Galois extensions of commutative rings, including several equivalent definitions, a Galois correspondence and a seven term exact sequence, involving Galois cohomology groups, Picard groups and the relative Brauer group. The sequence is seen as a common generalization of two classical facts on Galois extensions of fields: the Hilbert's Theorem 90 and the isomorphism of the relative Brauer group with the second Galois cohomology group. In a joint paper with M. Ferrero and A. Paques (2007) we defined Galois extensions in the context of partial group actions and obtained a Galois correspondence. It is natural to extend the Chase-Harrison-Rosenberg sequence for the case of partial Galois extensions of commutative rings. In a joint paper with A. Paques and H. Pinedo we built up homomorphisms, which are analogues of those in the Chase-Harrison-Rosenberg sequence. Now in a preprint with A. Paques, H. Pinedo and I. Rocha we prove that the sequence is exact. We shall recall the ingredients of the sequence and discuss some ideas used in the proof of the exactness.

Lecturer	Title	Date	Time	Room
Tran Giang Nam (Institute of Mathematics, VAST, Vietnam)	Leavitt path algebras - Something for everyone: algebra, analysis, graph theory (part 2)	April 30, 2019	14:30-15:30	242-A
Abstract: The rings studied by students in most first-year algebra courses have the "Invariant Basis Number'' property: for every pair of positive integers m and n, if the free left R-modules R^{m} and R^{n} are isomorphic, then m=n. For instance, the IBN property in the context of fields is simply the statement that any two bases of a vector space must have the same cardinality. Similarly, the IBN property for the ring of integers is a consequence of the Fundamental Theorem for Finitely Generated Abelian Groups. In important work completed in the early 1960's, William G. Leavitt produced a specific, universal collection of algebras which fail to have IBN. While these algebras were initially viewed as interesting but not so "main stream'', these now-so-called Leavitt algebras currently play a central, fundamental role in numerous lines of research in both algebra and analysis. More generally, from any directed graph E and any field K one can build the Leavitt path algebra L_K(E). In particular, the Leavitt algebras arise in this more general context as the algebras corresponding to the graphs consisting of a single vertex. The Leavitt path algebras were first defined in 2004; as of 2019 the subject is currently experiencing a seemingly constant opening of new lines of investigation, and the significant advancement of existing lines. I will give an overview of some of the work on Leavitt path algebras which has occurred in their fifteen years of existence, as well as mention some of the future directions and open questions in the subject. There should be something for everyone in this presentation, including and especially algebraists, analysts, and graph theorists.  We will also present an elementary  number theoretic observation which provides the foundation for one of the main results in Leavitt path algebras, a result which has had a number of important applications, including one in the theory of simple groups. The talk will be aimed at a general audience; for most of the presentation, a basic course in rings and modules will provide more-than-adequate background.

Lecturer	Title	Date	Time	Room
Tran Giang Nam (Institute of Mathematics, VAST, Vietnam)	Leavitt path algebras - Something for everyone: algebra, analysis, graph theory	April 23, 2019	14:30-15:30	242-A
Abstract: The rings studied by students in most first-year algebra courses have the "Invariant Basis Number'' property: for every pair of positive integers m and n, if the free left R-modules R^{m} and R^{n} are isomorphic, then m=n. For instance, the IBN property in the context of fields is simply the statement that any two bases of a vector space must have the same cardinality. Similarly, the IBN property for the ring of integers is a consequence of the Fundamental Theorem for Finitely Generated Abelian Groups. In important work completed in the early 1960's, William G. Leavitt produced a specific, universal collection of algebras which fail to have IBN. While these algebras were initially viewed as interesting but not so "main stream'', these now-so-called Leavitt algebras currently play a central, fundamental role in numerous lines of research in both algebra and analysis. More generally, from any directed graph E and any field K one can build the Leavitt path algebra L_K(E). In particular, the Leavitt algebras arise in this more general context as the algebras corresponding to the graphs consisting of a single vertex. The Leavitt path algebras were first defined in 2004; as of 2019 the subject is currently experiencing a seemingly constant opening of new lines of investigation, and the significant advancement of existing lines. I will give an overview of some of the work on Leavitt path algebras which has occurred in their fifteen years of existence, as well as mention some of the future directions and open questions in the subject. There should be something for everyone in this presentation, including and especially algebraists, analysts, and graph theorists.  We will also present an elementary  number theoretic observation which provides the foundation for one of the main results in Leavitt path algebras, a result which has had a number of important applications, including one in the theory of simple groups. The talk will be aimed at a general audience; for most of the presentation, a basic course in rings and modules will provide more-than-adequate background.

Lecturer	Title	Date	Time	Room
Daniel Kawai (IME-USP)	Graded division rings	April 9, 2019	14:30-15:30	242-A
Abstract: In ring theory, we often study properties about ring epimorphisms from a given ring R to division rings, that we call epic R-division rings. Paul Cohn discovered a correspondence between epic R-division rings and some sets of square matrices that he called prime matrix ideals. Moreover, he obtained a criterion for the existence of an epic R-division ring, and a criterion for R to be embeddable in a division ring. Given a ring R, Cohn considered a kind of morphism of R-division rings that he called specialization, and the existence of a specialization was shown to be equivalent to the property that every matrix over R that becomes invertible in L becomes invertible in K, and it was shown equivalent to the property that every rational relation over R that is valid in K is valid in L. Therefore, he also studied some important class of rings R there is an initial epic R-division ring, such as Sylvester domains, and pseudo-Sylvester domains, firs and semifirs. Moreover, if G is a free group and K is a division ring, the group ring K[G] is a semifir. We obtained a generalization of all these results in group graded rings, and we will present them.

Lecturer	Title	Date	Time	Room
Antonio Giambruno (Universitá di Palermo, Italia)	Polynomial identities and star-fundamental algebras	March 26, 2019	14:30-15:30	242-A
Abstract: In the theory of polynomial identities in characteristic zero developed by Kemer an important role is played by the so-called fundamental algebras. These are finite dimensional algebras defined in terms of multialternating polynomials and one of the main features is that any finite dimensional algebra has the same identities as a finite direct sum of fundamental algebras. In the last years we have developed together with La Mattina e Polcino Milies a theory of fundamental algebras in the setting algebras with involution. Among other results we are able to determine the polynomial factor in the asymptotics of the star-codimensions for any star-fundamental algebra.

Lecturer	Title	Date	Time	Room
Makar Plakhotnyk (IME-USP)	Combinatorial automorphisms of quasi semi metrics	March 19, 2019	14:30-15:30	242-A
Abstract: Quasi semi metrics on a finite set can be represented by square n\times n real non-negative matrices A = (\alpha_{pq}) such that \alpha_{ii}=0 for all i and \alpha_{ij} + \alpha_{jk}\geq \alpha_{ik} for all i, j, k. If we assume that the entries \alpha_{ij} are integers we obtain the definition of a non-negative exponent matrix. Exponent matrices are crucial ingredients of tiled orders and also related to semi maximal rings. One of the interesting features of the non-negative exponent matrices is that they form a max-plus algebra. Since the set of all quasi semi metrics forms a polyhedral cone, it is reasonable to consider their combinatorial automorphisms. We recall that a combinatorial automorphism of a polyhedral cone is an invertible map on the set of its faces, which preserves inclusion. Using some earlier results on exponent matrices, we prove that each combinatorial automorphism of the polyhedral cone of the quasi semi metrics is induced by a max-plus automorphism of the exponent matrices. This is a joint work with prof. M. Dokuchaev, which is also motivated by our interaction with prof. A. Mandel.

Lecturer	Title	Date	Time	Room
Cesar Polcino Milies (IME-USP)	Essential idempotents in group algebras and coding theory	March 12, 2019	14:30-15:30	242-A
Abstract: A primitive idempotent e in a semisimple group algebra F_qG is called essential if e\widehat{H} = 0 for every normal subgroup H of G. We give some criteria to decide when a primitive idempotent is essential, and compute the number of essential idempotents in cyclic group algebras. Then, we find a relation among essential idempotents in different group algebras. Along the way, we explore their significance for coding theory.

Lecturer	Title	Date	Time	Room
Itailma Rocha (UFCG)	Uma sequência exata relacionada a uma extensão de anéis e uma representação parcial.	February 26, 2019	14:30-15:30	249-A
Abstract: Para uma extensão de Galois de anéis comutativos, Chase-Harrison-Rosenberg construíram uma sequência exata de sete termos que envolve o grupo de Picard, o grupo de Brauer relativo e grupos de cohomologias. Essa sequência é vista como uma generalização de dois fatos importantes da teoria galoisiana de corpos, a saber, o Teorema 90 de Hilbert e o isomorfismo de grupo de Brauer relativo com o segundo grupo de cohomologia. A sequência foi generalizada por Miyashita para o contexto de anéis não comutativos com unidade. Mais tarde, El Kaoutit e Gómez-Torrencillas generalizaram o resultado de Miyashita para uma extensão de anéis não comutativos e não unitais, apenas com um conjunto de unidades locais. A sequência de Chase-Harrison-Rosenberg também foi considerada para ações parciais por Dokuchaev, Paques e Pinedo, que construíram uma versão para uma extensão de Galois parcial de anéis comutativos. Nesta tese, elaboramos uma versão da sequência no contexto de ações parciais para uma extensão de anéis não comutativos com unidade. A sequência apresentada aqui generaliza a sequência dada por Miyashita.